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Unformatted text preview: MAS 213: Linear Algebra II. Problem list for Week #1. Tutorial on 7th September. This week’s topics: • The definition of a vector space. • Examples of vector spaces. • Subspaces. Tutorial problems: Problem 1: (Problem 1.2.13, 1.2.17-19 in [FIS].) Let V denote the set of ordered pairs of real numbers. In the following 4 parts, we are going to attempt to put an R-vector space structure on V by making different definitions of vector addition and scalar multiplication. For each of the following examples, either check that all the axioms hold, or find an axiom which fails to hold. (To show it fails to hold you should find some particular vectors/scalars where the equation fails.) 1. Define the operations by ( a 1 , a 2 ) + ( b 1 , b 2 ) = ( a 1 + b 1 , a 2 b 2 ) and c ( a 1 , a 2 ) = ( ca 1 , a 2 ). 2. Define vector addition in the usual (i.e. component-wise) way and scalar multiplication by c ( a 1 , a 2 ) = ( a 1 , 0). 3. Define operations by ( a 1 , a 2 )+( b 1 , b 2 ) = ( a 1 +2 b 1 , a 2 +3 b 2 ) and scalar multiplication in the usual (i.e. component-wise) way....
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This note was uploaded on 12/08/2010 for the course SPMS MAS213 taught by Professor Andrewkricker during the Fall '10 term at Nanyang Technological University.
- Fall '10